Express of a statement that have universal quantifiers

[pdaogu]

I am beginner to Advanced Math. I have a assignment that:

Q. Let C(x, y) mean that student x is enrolled in class y, where the domain for x consists of all students in your school and the domain for y consists of all classes being given at your school. Express of these statements by a simple sentence.

I have a statement:

. . . . .\(\displaystyle \exists x\, \exists y\, \forall z\, \bigg((x\, \neq\, y)\, \wedge\, \big(C(x,\, z)\, \rightarrow\, C(y,\, z)\big)\bigg)\)

And the answer is written in that book:

A. There exist at least two students such that if one is enrolled in every courses, then the other

However, I think it should be:

B. There exist at least two different students such that every courses which the one enrolled, then the other.

Are that two sentences, A and B, different? Which sentence is true? Please explain for me.

 

[stapel]

I am beginner to Advanced Math. I have a assignment that:

Q. Let C(x, y) mean that student x is enrolled in class y, where the domain for x consists of all students in your school and the domain for y consists of all classes being given at your school. Express of these statements by a simple sentence.

I have a statement:

. . . . .\(\displaystyle \exists x\, \exists y\, \forall z\, \bigg((x\, \neq\, y)\, \wedge\, \big(C(x,\, z)\, \rightarrow\, C(y,\, z)\big)\bigg)\)

And the answer is written in that book:

A. There exist at least two students such that if one is enrolled in every courses, then the other

However, I think it should be:

B. There exist at least two different students such that every courses which the one enrolled, then the other.

Are that two sentences, A and B, different? Which sentence is true? Please explain for me.

I would say that your sentence is the better one. The first one reads as though the author is not a native speaker of English, so I may be misunderstanding the meaning. But I don't see the "for all z" to mean that "the student x is enrolled in all classes z, and thus so also is student y". I see it as meaning something more along the lines that, "for any class z in which student x is enrolled, student y is also enrolled, and this if-then is true for any and all classes in which student x enrolls". And, of course, there exist two students, x and y, for whom this is true.